Isometric and Isoenergetic Everting Motions of Orientable and Nonorientable Bands

Abstract We present a framework for describing continuous, periodic everting motions of orientable and nonorientable bands, including those with knotted topologies, thereby extending the scope of eversion beyond simple hemispheres and cylindrical bands. The framework involves deriving a kinematic evolution equation and analytically constructing its general traveling wave solution, which captures periodic motions for both orientable and nonorientable bands, including knotted configurations. Using this solution, the configuration of a band at any instant during an eversion cycle is described through periodic or antiperiodic extensions of the initial data, depending on whether the band is orientable or nonorientable, respectively. Each motion is isometric, preserving distances between points, and isoenergetic, maintaining constant elastic bending energy throughout the motion. By introducing shape-preserving motions beyond rigid translations and rotations, these results expand the known classes of eversion and offer new insights into the design of soft robotic systems and adaptive materials. These findings may open up new avenues for applications in fields such as medical, aerospace, and architectural engineering, for which adaptability and multifunctionality are critical.